Convergence of the Fourier Transform from Fourier Analysis a discrete time non periodic signal.

1. Convergence of the Fourier
Transform
Group 5

2. Outline
 Fourier Transform
 Application Of fourier Transform
 Fourier Series
 Different between Fourier Transform and series
 Dirichlet Condition
 Fourier Analysis of discrete time
 Convergence of Fourier Transform

3. Fourier Transform
The Fourier transform simply states that that
the non periodic signals whose area under the
curve is finite can also be represented into
integrals of the sines and cosines after being
multiplied by a certain weight.

4. Application of Fourier Transform
Image compression
Filtering
Image analysis.

5. Fourier Series
Fourier series simply states that, periodic
signals can be represented into sum of sines
and cosines when multiplied with a certain
weight

6. Difference Between Fourier Transform and
series
Both Fourier series and Fourier transform are
given by Fourier , but the difference between
them is Fourier series is applied on periodic
signals and Fourier transform is applied for
non periodic signals

7. Dirichlet Condition
Signal should have finite number of mixima and
minima over any finite number
Signal should have number of discontinuities over
any finite interval
Signal should have be absolutely integrable

8. Fourier Analysis of discrete time

9. Fourier Transform
Uniform convergence

10. Absolutely Summable
Square Summable

11. Mean square Convergence

12. Example
Infinite Energy
Sequence of Equation

13. Infinite series Finite series

14. Convergence of the FT

15. Gibbs Phenomenon
The oscillatory behavior of the approximation Xn(W)
to the x(w) at the point of discontinuity of x(w) is
called Gibbs Phenomenon.

16. Thank You